 # is a Positive Number

Students in middle school experience the knowledge that the product of two negative numbers is a positive number.   The challenge in this "problem" is to delineate alternative demonstrations that may appeal to students' reasoning so as to understand "why" this is so.

Here are some suggestions for you to explore and develop.    This is sort of a "catalog" of a few strategies and most would need to be expanded to become a viable demonstration.

Perhaps it is open to debate as to whether alternatives are useful vs having one explanation and sticking to it.

## Numerical Patterns

5  x  -2 = -10
4  x  -2 =   -8
3  x  -2 =   -6
2  x  -2 =   -4
1  x  -2 =   -2
0  x  -2 =    0
-1  x  -2 =    2
-2  x  -2 =    4

This is a frequently used and powerful strategy for teachers to engage students in inductive reasoning.    The idea is to use sufficient and varied examples for students to identify patterns and form generalizations.    Multiple examples and discussion are needed.

## Use of commutative, associative, distributive, and additive inverse laws

For example  6 = 2  x  3

What is          2  x  3     +   (-2 )  x  3   +    (-2)   x   (-3)     ?

2  x  3     +   (-2 )  x  3   +    (-2)   x   (-3)
=  2  x  3   +   (-2)  x  [   3    +     ( - 3)     ]                     factor out  (-2)
= (2)  x   (3)    +   (-2)  x  [  0  ]                                      because  3 + (-3) = 0
= (2)  x   (3)     +      0
= (2)  x  (3)

But if we take the same  expression and factor out  (3) from the first two terms

2  x  3     +   (-2 )  x  3   +    (-2)   x   (-3)
= [  2    +   (-2)  ]  x 3    +  (-2) x (-3)                              factor out  3
= [  0  ] x (3)    +     (-2) x (-3)
=  0  +      (-2) x (-3)
=  (-2) x (-3)

So in the one case our expression is   2 x 3  and  in the other it is  (-2) x (-3).

So  (-2) x (-3)   is the same number as  2 x 3.

## Formalizing the above numerical strategy

Let   a   and   b  be any two positive real numbers.   Define   x    by

x = ab + (-a) (b)  + (-a)(-b)

On the one hand,     x = ab   +  (-a)[  (b)  +  (-b) ]                 { factor out   - a}
= ab  +   (-a)(0)
= ab  +  0
= ab

On the other hand  x  = [     a  +  (-a) ] (b)  +  (-a)(-b)           {factor out   b}
= 0  (b)  + (-a)(-b)
= 0   +  (-a)(-b)
= (-a)(-b)

So we have   x  =  ab    and    x = (-a)(-b).   Therefore the product of two negative numbers is  positive.

## Use of a number line model

1. Distance, direction, and time.    Movement along a road forward and backward in time is related to this.

2.  Reflection

These are two different approaches with the number line.   Describe how you would implement each of them.

## Use of applications

Think of debt as a negative number, saving as a positive number.

Suppose your employer deducts \$120 per month from your paycheck for
health insurance.       In 6 months the deduction (debt) is  6 x (-\$120) =  -\$720

Now suppose your employer removes this deduction as a benefit to your employment.  “Removes”   is a negative so in it is a gain   for you and  in six months
-6  x (-\$120) amounts to  \$720.        Here a  negative  times a negative is a positive.

Consider other (perhaps clearer)  "applications."

Elevators

Football

Snooker scores

Time and distance, each positive and negative.

## Another mathematical explanation

A negative number is just a positive number multiplied by   - 1.

So the product of two negative numbers   (-a)(-b) =  (-1)(-1)ab where ab is positive

Now what is   (-1)(-1)?

We agree that   1 +  (-1)   = 0        and     (-1) x 0 = 0
Now,   0 = (- 1) x 0
=  (-1) (1 + (-1) )
= (-1)(1)  +  (-1)(-1)

Since this sum is    0    and   (-1)(1) =  -1,  the    (-1)(-1)  must be 1.

ALTERNATIVE:   If we assumed the opposite that

(-1)(-1) = -1,

then our sum would  be  -2 and this is a contradiction.   Therefore   (-1)(-1) =   1.

## Interpret Multiplication as repeated addition.

3 x 2 =   2 + 2 + 2 = 6

3 x (-2) = (-2) + (-2) + ( -2)  = - 6

(-3) x 2 =  2 subtracted 3 times = -2  -2  - 2 = -6

(-3) x (-2) = -2 subtracted 3 times = 2 + 2 + 2

This is, in fact, a frequently used strategy for elementary and middle school teachers.   Think about the accompanying explanation to help students understand that "subtracting  -2" is the same as adding positive 2.

## Use the additive inverse.

The additive inverse of  +3  is  -3;  the additive inverse of -3 is + 3

Therefore, (-3) x (-2) must be the additive inverse of   3 x (-2) or  (-3) x 2.

But each of  3 x (-2)   and (-3) x 2  is the additve inverse of  3 x 2.

## Analogy with the "logic" of double negation.

Advocates of this strategy use the logic of statement  like "That cat is not not cool." to mean "That cat is cool."    By analogy, they develop that a negative times a negative is positive.

## The Light Switch

Think of the negative sign as a light switch.   If the light is on and you flip the switch once, the light turns off.  You flip it again and the light turns on again.   Negative followed by a negative gets a positive.

## Videotape of a water container filling and emptying

Consider a videotape of a water container being filled and emptied.  When the video is running forward as the container fills, it shows the water level increasing: Positive/positive leads to positive.

When the the videotape is running forward as the container empties, it shows the water level decreasing:    Positive/negative leads to negative.

When the videotape is running backward, the filling appears to decrease the level of the water:    Negative/positive leads to negative.

When the videotape is running backward,  the emptying appears to increase the level of the water:     Negative/negative leads to positive.

[Note:   I tried to find an example of such a videotape on the internet but was unsuccessful.  I know they exist somewhere.   If anyone finds one, give me the URL.]

## Reference:

Peterson, John C.  (1972).   Fourteen different strategies for multiplication of integers or why  (-1)(-1) = 1. The Aritimetic Teacher, 19(5), 396-403.

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